Higher category theory

Higher category theory is an advanced area of mathematics that generalizes the concepts of category theory by enriching the structure of categories to include "higher" morphisms. In basic category theory, you have objects and morphisms (arrows) between those objects. Higher category theory extends this by allowing for morphisms between morphisms, known as 2-morphisms, and even higher levels of morphisms, creating a hierarchy of structures.

A bicategory is a generalization of the concept of a category in category theory. While a category consists of objects and morphisms (arrows) between those objects, a bicategory includes not only objects and morphisms but also "2-morphisms" (which can be thought of as arrows between arrows). Here are the key features of a bicategory: 1. **Objects**: Just like in categories, a bicategory has objects.

In mathematics, specifically in the context of algebra and set theory, the term "conglomerate" does not have a widely recognized definition that is analogous to concepts like group, ring, or field. It may refer to different concepts depending on the discipline or context in which it is used. 1. **General Context**: A "conglomerate" can sometimes refer to a collection of objects or entities that are grouped together based on some common property, much like how a set is defined.

Extranatural transformation refers to a concept in the field of mathematics, particularly in category theory and algebraic topology. While it is not as commonly discussed as some other concepts, the idea generally pertains to the transformation of objects or morphisms within a specific framework that extends beyond traditional natural transformations. In category theory, a **natural transformation** is a way of transforming one functor into another while preserving the structure of the categories involved.

Higher Topos Theory is a branch of mathematical logic and category theory that extends the concepts of topos theory to higher-dimensional or higher categorical settings. At its core, topos theory studies topoi (plural of topos), which are categories that behave similarly to the category of sets, allowing for a rich interplay between algebra, geometry, and logic.

The N-category number is a mathematical concept originating from the field of category theory, particularly in the study of higher categories and homotopy theory. It effectively captures the notion of "categorical" structures that extend beyond the traditional notion of categories, which are typically composed of objects and morphisms (arrows) between those objects.

An **N-monoid** is a concept in the field of algebra, specifically in the study of algebraic structures known as monoids. A monoid is a set equipped with an associative binary operation and an identity element. 1. **Basic Definition of a Monoid**: - A set \( M \) along with a binary operation \( \cdot: M \times M \to M \) (often written simply as juxtaposition, i.e.

A **stable ∞-category** is a concept from higher category theory that arises in the study of derived categories and stable homotopy theory. It is a type of ∞-category (a category made up of higher-dimensional morphisms) that possesses certain stability properties, much like how stable homotopy categories have homotopy classes of maps that behave well under suspension.

A **strict 2-category** is a generalization of a category that allows for a richer structure by incorporating not just objects and morphisms (arrows) between them, but also higher-dimensional morphisms called 2-morphisms (or transformations) between morphisms. In a strict 2-category, all the structural relationships between objects, morphisms, and 2-morphisms are explicitly defined and obey strict associativity and identity laws.

A string diagram is a visual representation used in various fields, most prominently in mathematics and physics, particularly in category theory and string theory. The term may be interpreted in different contexts, but here are the two primary uses: 1. **String Diagrams in Category Theory**: - In category theory, string diagrams are a way to visualize morphisms (arrows) and objects (points) within a category.

A tetracategory is a type of higher categorical structure that extends the concept of categories and higher categories. In general, a **category** consists of objects and morphisms (arrows) between those objects that can be composed. A **2-category** extends this idea by allowing morphisms between morphisms, known as 2-morphisms.

A tricategory is a generalization of a category in the context of higher category theory. While a category consists of objects and morphisms (arrows) between those objects, a tricategory extends this idea to include not just objects and morphisms, but also a second layer of structure called 2-morphisms, and a third layer called 3-morphisms.

Weak \( n \)-categories are a generalization of the concept of \( n \)-categories in the field of higher category theory. In traditional category theory, a category consists of objects and morphisms between those objects, satisfying certain axioms. As we move to higher dimensions, such as \( 2 \)-categories or \( 3 \)-categories, we introduce higher-dimensional morphisms (or "cells"), leading to more complex structures.

An ∞-topos is a concept in higher category theory that generalizes the notion of a topos, which originates from category theory and algebraic topology. In classical terms, a topos can be considered as a category that behaves like the category of sheaves on a topological space, possessing certain properties such as limits, colimits, exponentials, and a subobject classifier.